Research Papers

Autofrettage and Shakedown Analyses of an Internally Pressurized Thick-Walled Cylinder Based on Strain Gradient Plasticity Solutions

[+] Author and Article Information
X.-L. Gao

ASME Fellow
Department of Mechanical Engineering,
Southern Methodist University,
P.O. Box 750337,
Dallas, TX 75275-0337
e-mail: xlgao@smu.edu

J.-F. Wen

School of Mechanical and Power Engineering,
East China University of Science
and Technology,
130 Meilong Road,
Shanghai 200237, China

F.-Z. Xuan, S.-T. Tu

School of Mechanical and Power Engineering,
East China University of Science
and Technology,
130 Meilong Road,
Shanghai 200237, China

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 23, 2014; final manuscript received February 10, 2015; published online February 27, 2015. Assoc. Editor: George Kardomateas.

J. Appl. Mech 82(4), 041010 (Apr 01, 2015) (12 pages) Paper No: JAM-14-1537; doi: 10.1115/1.4029798 History: Received November 23, 2014; Revised February 10, 2015; Online February 27, 2015

Two closed-form solutions for an internally pressurized thick-walled cylinder of an elastic linear-hardening material and of an elastic power-law hardening material are first obtained using a strain gradient plasticity theory, a unified yield criterion, and Hencky's deformation theory. The strain gradient plasticity theory contains a microstructure-dependent length-scale parameter and can capture size effects observed at the micron scale. The unified yield criterion includes the intermediate principal stress and recovers the Tresca, von Mises, and twin shear yield criteria as special cases. An autofrettage analysis is then performed by using the two new solutions, which leads to the analytical determination of the elastic and plastic limiting pressures, the residual stress field, and the stress field induced by an operating pressure for each strain-hardening cylinder. This is followed by a shakedown analysis of the autofrettaged thick-walled cylinders, which results in analytical formulas for reverse yielding and elastic reloading shakedown limits. The newly obtained solutions and formulas include their classical plasticity-based counterparts as limiting cases. To quantitatively illustrate the new formulas derived, a parametric study is conducted. The numerical results reveal that the shakedown limit (as the upper bound of the autofrettage pressure) increases with the diameter ratio and with the strain hardening level. It is also found that the Tresca yield criterion gives the lowest value and the twin shear yield criterion leads to the highest value, while the von Mises yield criterion results in the intermediate value of the shakedown limit. In addition, it is observed that the shakedown limit based on the current strain gradient plasticity solutions increases with the decrease of the inner radius when the cylinder inner radius is sufficiently small, but it approaches that (a constant value independent of the inner radius) based on the classical plasticity solution when the inner radius becomes large. This predicted size (strengthening) effect at the micron scale agrees with the general trends observed experimentally.

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Bland, D. R., 1956, “Elastoplastic Thick-Walled Tubes of Work-Hardening Materials Subject to Internal and External Pressures and to Temperature Gradients,” J. Mech. Phys. Solids, 4(4), pp. 209–229. [CrossRef]
Durban, D., 1979, “Large Strain Solution for Pressurized Elasto/Plastic Tubes,” ASME J. Appl. Mech., 46(1), pp. 228–230. [CrossRef]
Durban, D., and Kubi, M., 1992, “A General Solution for the Pressurized Elastoplastic Tube,” ASME J. Appl. Mech., 59(1), pp. 20–26. [CrossRef]
Gao, X.-L., 1992, “An Exact Elasto-Plastic Solution for an Open-Ended Thick-Walled Cylinder of a Strain-Hardening Material,” Int. J. Pressure Vessels Piping, 52(1), pp. 129–144. [CrossRef]
Gao, X.-L., 1993, “An Exact Elasto-Plastic Solution for a Closed-End Thick-Walled Cylinder of Elastic Linear-Hardening Material With Large Strains,” Int. J. Pressure Vessels Piping, 56(3), pp. 331–350. [CrossRef]
Perry, J., and Aboudi, J., 2003, “Elasto-Plastic Stresses in Thick Walled Cylinders,” ASME J. Pressure Vessel Technol., 125(3), pp. 248–252. [CrossRef]
Zhao, W., Seshadri, R., and Dubey, R. N., 2003, “On Thick-Walled Cylinder Under Internal Pressure,” ASME J. Pressure Vessel Technol., 125(3), pp. 267–273. [CrossRef]
Davidson, T. E., Kendall, D. P., and Reiner, A. N., 1963, “Residual Stresses in Thick-Walled Cylinders Resulting From Mechanically Induced Overstrain,” Exp. Mech., 3(11), pp. 253–262. [CrossRef]
Chen, P. C. T., 1986, “The Bauschinger and Hardening Effect on Residual Stresses in an Autofrettaged Thick-Walled Cylinder,” ASME J. Pressure Vessel Technol., 108(1), pp. 108–112. [CrossRef]
Liu, X. S., and Xu, B. Y., 1990, “On the Shakedown Analysis of Thick-Walled Cylindrical Tube,” Shanghai Mech., 11(4), pp. 1–9.
Jiang, W., 1992, “The Elastic-Plastic Analysis of Tubes—III: Shakedown Analysis,” ASME J. Pressure Vessel Technol., 114(2), pp. 229–235. [CrossRef]
Avitzur, B., 1994, “Autofrettage–Stress Distribution Under Load and Retained Stresses After Depressurization,” Int. J. Pressure Vessels Piping, 57(3), pp. 271–287. [CrossRef]
Lazzarin, P., and Livieri, P., 1997, “Different Solutions for Stress and Strain Fields in Autofrettaged Thick-Walled Cylinders,” Int. J. Pressure Vessels Piping, 71(3), pp. 231–238. [CrossRef]
Xu, S.-Q., and Yu, M.-H., 2005, “Shakedown Analysis of Thick-Walled Cylinders Subjected to Internal Pressure With the Unified Strength Criterion,” Int. J. Pressure Vessels Piping, 82(9), pp. 706–712. [CrossRef]
Huang, X. P., 2005, “A General Autofrettage Model of a Thick-Walled Cylinder Based on Tensile-Compressive Stress-Strain Curve of a Material,” J. Strain Anal. Eng. Des., 40(6), pp. 599–607. [CrossRef]
Yu, M.-H., Ma, G.-W., Qiang, H.-F., and Zhang, Y.-Q., 2006, Generalized Plasticity, Springer, Berlin.
Korsunsky, A. M., 2007, “Residual Elastic Strains in Autofrettaged Tubes: Elastic–Ideally Plastic Model Analysis,” ASME J. Eng. Mater. Technol., 129(1), pp. 77–81. [CrossRef]
Hojjati, M. H., and Hassani, A., 2007, “Theoretical and Finite-Element Modeling of Autofrettage Process in Strain-Hardening Thick-Walled Cylinders,” Int. J. Pressure Vessels Piping, 84(5), pp. 310–319. [CrossRef]
Zheng, X.-T., and Xuan, F.-Z., 2010, “Investigation on Autofrettage and Safety of the Thick-Walled Cylinder Under Thermo-Mechanical Loadings,” Chin. J. Mech. Eng., 46(16), pp. 156–161. [CrossRef]
Zheng, X.-T., and Xuan, F.-Z., 2011, “Autofrettage and Shakedown Analysis of Strain-Hardening Cylinders Under Thermo-Mechanical Loadings,” J. Strain Anal. Eng. Des., 46(1), pp. 45–55. [CrossRef]
Maugin, G. A., 2011, “A Historical Perspective of Generalized Continuum Mechanics,” Mechanics of Generalized Continua, H.Altenbach, G. A.Maugin, and V.Erofeev, eds., Springer, Berlin, pp. 3–19.
Gao, X.-L., 2003, “Elasto-Plastic Analysis of an Internally Pressurized Thick-Walled Cylinder Using a Strain Gradient Plasticity Theory,” Int. J. Solids Struct., 40(23), pp. 6445–6455. [CrossRef]
Gao, X.-L., 2007, “Strain Gradient Plasticity Solution for an Internally Pressurized Thick-Walled Cylinder of an Elastic Linear-Hardening Material,” Z. Angew. Math. Phys., 58(1), pp. 161–173. [CrossRef]
Hutchinson, J. W., 2000, “Plasticity at the Micron Scale,” Int. J. Solids Struct., 37(1–2), pp. 225–238. [CrossRef]
Mühlhaus, H.-B., and Aifantis, E. C., 1991, “A Variational Principle for Gradient Plasticity,” Int. J. Solids Struct., 28(7), pp. 845–857. [CrossRef]
Fan, S. C., Yu, M.-H., and Yang, S. Y., 2001, “On the Unification of Yield Criteria,” ASME J. Appl. Mech., 68(2), pp. 341–343. [CrossRef]
Fleck, N. A., and Hutchinson, J. W., 1993, “A Phenomenological Theory for Strain Gradient Effects in Plasticity,” J. Mech. Phys. Solids, 41(12), pp. 1825–1857. [CrossRef]
Gao, H., Huang, Y., Nix, W. D., and Hutchinson, J. W., 1999, “Mechanism-Based Strain Gradient Plasticity—I. Theory,” J. Mech. Phys. Solids, 47(6), pp. 1239–1263. [CrossRef]
Huang, Y., Gao, H., Nix, W. D., and Hutchinson, J. W., 2000, “Mechanism-Based Strain Gradient Plasticity—II. Analysis,” J. Mech. Phys. Solids, 48(1), pp. 99–128. [CrossRef]
Chen, S. H., and Wang, T. C., 2000, “A New Hardening Law for Strain Gradient Plasticity,” Acta Mater., 48(16), pp. 3997–4005. [CrossRef]
Gudmundson, P., 2004, “A Unified Treatment of Strain Gradient Plasticity,” J. Mech. Phys. Solids, 52(6), pp. 1379–1406. [CrossRef]
Fleck, N. A., and Willis, J. R., 2009, “A Mathematical Basis for Strain-Gradient Plasticity Theory—Part I: Scalar Plastic Multiplier,” J. Mech. Phys. Solids, 57(1), pp. 161–177. [CrossRef]
Gurtin, M. E., and Anand, L., 2009, “Thermodynamics Applied to Gradient Theories Involving the Accumulated Plastic Strain: The Theories of Aifantis and Fleck and Hutchinson and Their Generalization,” J. Mech. Phys. Solids, 57(3), pp. 405–421. [CrossRef]
Gupta, A., Steigmann, D. J., and Stölken, J. S., 2011, “Aspects of the Phenomenological Theory of Elastic-Plastic Deformation,” J. Elast., 104(1–2), pp. 249–266. [CrossRef]
Niordson, C. F., and Hutchinson, J. W., 2011, “Basic Strain Gradient Plasticity Theories With Application to Constrained Film Deformation,” J. Mech. Mater. Struct., 6(1–4), pp. 395–416. [CrossRef]
Krishnan, J., and Steigmann, D. J., 2014, “A Polyconvex Formulation of Isotropic Elastoplasticity Theory,” IMA J. Appl. Math., 79(5), pp. 722–738. [CrossRef]
Wei, Y., and Hutchinson, J. W., 2003, “Hardness Trends in Micron Scale Indentation,” J. Mech. Phys. Solids, 51(11), pp. 2037–2056. [CrossRef]
Gao, X.-L., 2006, “An Expanding Cavity Model Incorporating Strain-Hardening and Indentation Size Effects,” Int. J. Solids Struct., 43(21), pp. 6615–6629. [CrossRef]
Coleman, B. D., and Hodgdon, M. L., 1985, “On Shear Bands in Ductile Materials,” Arch. Ration. Mech. Anal., 90(3), pp. 219–247. [CrossRef]
Gao, X.-L., 2002, “Analytical Solution of a Borehole Problem Using Strain Gradient Plasticity,” ASME J. Eng. Mater. Technol., 124(3), pp. 365–370. [CrossRef]
Yu, M.-H., 2002, “Advances in Strength Theories for Materials Under Complex Stress State in the 20th Century,” ASME Appl. Mech. Rev., 55(3), pp. 169–218. [CrossRef]
Yu, M.-H., 1983, “Twin Shear Stress Yield Criterion,” Int. J. Mech. Sci., 25(1), pp. 71–74. [CrossRef]
Little, R. W., 1973, Elasticity, Prentice-Hall, Englewood Cliffs, NJ.
Hill, R., 1950, The Mathematical Theory of Plasticity, Oxford University Press, Oxford, UK.
Sowerby, R., and Uko, D. K., 1979, “A Review of Certain Aspects of the Bauschinger Effect in Metals,” Mater. Sci. Eng., 41(1), pp. 43–58. [CrossRef]
Bate, P. S., and Wilson, D. V., 1986, “Analysis of the Bauschinger Effect,” Acta Metall., 34(6), pp. 1097–1105. [CrossRef]
Parker, A. P., Underwood, J. H., and Kendall, D. P., 1999, “Bauschinger Effect Design Procedures for Autofrettaged Tubes Including Material Removal and Sachs' Method,” ASME J. Pressure Vessel Technol., 121(4), pp. 430–437. [CrossRef]
Livieri, P., and Lazzarin, P., 2002, “Autofrettaged Cylindrical Vessels and Bauschinger Effect: An Analytical Frame for Evaluating Residual Stress Distributions,” ASME J. Pressure Vessel Technol., 124(1), pp. 38–46. [CrossRef]
de Swardt, R. R., 2006, “Material Models for the Finite Element Analysis of Materials Exhibiting a Pronounced Bauschinger Effect,” ASME J. Pressure Vessel Technol., 128(2), pp. 190–195. [CrossRef]
Chen, H. F., 2009, “Lower and Upper Bound Shakedown Analysis of Structures With Temperature-Dependent Yield Stress,” ASME J. Pressure Vessel Technol., 132(1), p. 011202. [CrossRef]
Chen, H. F., and Ponter, A. R. S., 2001, “Shakedown and Limit Analyses for 3-D Structures Using the Linear Matching Method,” Int. J. Pressure Vessels Piping, 78(6), pp. 443–451. [CrossRef]
Polizzotto, C., 1993, “On the Conditions to Prevent Plastic Shakedown of Structures: Part II—The Plastic Shakedown Limit Load,” ASME J. Appl. Mech., 60(1), pp. 20–25. [CrossRef]
Maier, G., 2001, “On Some Issues in Shakedown Analysis,” ASME J. Appl. Mech., 68(5), pp. 799–808. [CrossRef]
Zhu, H. T., Zbib, H. M., and Aifantis, E. C., 1997, “Strain Gradients and Continuum Modeling of Size Effect in Metal Matrix Composites,” Acta Mech., 121(1–4), pp. 165–176. [CrossRef]


Grahic Jump Location
Fig. 1

Cylinder cross section

Grahic Jump Location
Fig. 2

Variation of piS with ro/ri: (a) elastic linear-hardening cylinders and (b) elastic power-law hardening cylinders

Grahic Jump Location
Fig. 3

Variation of piS with the inner radius ri: (a) elastic linear-hardening cylinders and (b) elastic power-law hardening cylinders



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